Bullet voting explained

Bullet,^[1] single-shot,^[2] or plump voting^[3] is when a voter supports only a single candidate, typically to show strong support for a single favorite.

Every voting method that does not satisfy either later-no-harm (most methods) or monotonicity (such as instant-runoff voting) will encourage bullet voting or truncation in some situations.^{[4] [5]}

In systems that fail later-no-harm, voters who feel strongly about their favorite candidate can use bullet voting to maximize the chances their favorite candidate will be elected, at the cost of reducing the chances that one of their later preferences will win.^{[6] [7]}

In non-participatory systems (such as instant-runoff), voters can sometimes strategically bullet-vote to hide their support for additional candidates; this strategy works because such systems can cause candidates to lose when they receive too much support from voters.

Setups where voters may benefit from truncating their ballots are sometimes called a truncation paradox or Burr dilemma. This name comes from Aaron Burr, who tied with Thomas Jefferson in the Electoral College during the 1800 election after one Jefferson-Burr elector forgot to truncate their ballot to exclude Burr, almost causing a constitutional crisis.^{[8] [9]}

In systems like cumulative voting, bullet voting is actively encouraged as a way for minority groups to achieve proportional representation, by allowing small groups to concentrate all their support on one candidate and win at least one seat on a city council.^[10] During the Jim Crow era, municipalities often banned or disparaged bullet voting in an attempt to prevent black voters from being able to achieve representation on city councils, creating a stigma that in some cases lasts to the present day.

Single winner elections

Plurality

First-preference plurality is usually modeled as a ranked voting system where voters can rank as many candidates as they like, and the candidate with the most first-preference votes wins. As a result, plurality is "immune" to bullet voting or truncation as a strategy, but only by making every vote equivalent to a bullet vote.

Instant-runoff

Contrary to a common misconception, later-no-harm systems like instant-runoff are not immune to truncation, unless they satisfy the participation criterion as well; because instant-runoff fails participation, it encourages bullet voting or truncation in some circumstances.

Graham-Squire and McCune note that instant-runoff can suffer from an especially severe kind of strategic truncation, stronger than bullet voting, where voters cannot safely rank any candidates at all; such a situation is called a no-show paradox. A 2021 study found roughly 32% of voters under instant-runoff cast bullet-votes, although it suggested this had more to do with convenience than with strategic incentives.^[11]

Cardinal voting

By contrast, in approval and score voting, bullet voting can be fairly common for voters who only feel one candidate adequately represents them. However, because approval satisfies no favorite betrayal, such voting is not deceptive (in other words, it accurately reflects a voter's honest ordering of candidates). In general, the optimal strategy for an approval voter is to approve of all above-average candidates, i.e. all candidates whose quality is higher than the expected value of the winner.

Traditional Bucklin voting was infamous for its high vulnerability to bullet voting, due to its use of ranked ballots;^[12] however, Balinski and Laraki showed in their study of highest median rules that this can be fixed by using rated ballots, which allow voters to skip ratings to show weak support for a candidate.

Multiple winner elections

Votes
Approval voting Range voting Borda voting	Point scores
Plurality-at-large voting	N
Limited voting	N-1 N-2 ...
Single non-transferable vote (Whole vote)	1
Instant-runoff voting (Whole vote)	1
Cumulative voting (Explicit divided vote)	1
Single transferable vote (Implicit divided vote)	1

Multiple votes are often allowed in elections with more than one winner. Bullet voting can help a first choice be elected, depending on the system:

Multiple non-transferable vote methods

• Approval voting used in a multi-winner election works the same way as Plurality-at-large but allows more votes than winners. This results in a body that is less representative than a body elected under a proportional voting method but would still have the same ideological center as the population. 100% bullet voting under Approval Voting in a multi-winner election is unlikely, as voters are incentivized to vote for acceptable moderates in addition to their favorite candidates to avoid being locked out of the election entirely.^[13]
  • Range voting is a generalization of Approval voting where voters can give gradations of support for each candidate. Here bullet voting refers to providing 100% support for one candidate and 0% for all other candidates, just like Approval bullet voting.
  • Borda voting assign multiple votes based on ranked ballots, like three votes for the first, two for the second, and one for the third choice. This encourages minority voters to bullet vote (not using all the rankings). If voters are required to rank all the candidates, it further encourages voters to (insincerely) bury strongest rivals at the lowest rankings.
• Plurality-at-large voting (Bloc-voting) allows up to N votes for elections with N winner elections. In this system, a voter who prefers a single candidate and is concerned his candidate will lose has a strong incentive to bullet vote to avoid a second choice helping to eliminate the first choice.^[14] A united majority of voters in plurality-at-large can control all the winners despite any strategic bullet voting by a united minority.

Limited-Vote Systems

• A single nontransferable vote limits everyone to one vote, effectively making bullet voting mandatory, minimizing the power of a majority of voters to pick all the winners, and can work well if there are only a few more candidates as winners.^[15]
• Cumulative voting allows up to N votes for N winner elections which can be distributed between multiple candidates or all given to one candidate. Effectively, this is one vote which can be fractionally divided among more than one candidate. This removes any penalty to bullet voters, who support a single candidate, and it enables the possibility of a united minority to elect at least one winner despite a united majority voting for all other candidates.
• Single transferable vote removes the incentive for sincere strategic bullet voting, since lower rankings are only used later (after higher choices have been elected or eliminated). However, it creates an incentive for insincere bullet voting in situations where adding additional preferences would hurt later candidates; in some situations, adding a later preference can cause that later-preference candidate to lose.

Burr dilemma

The Burr dilemma is a particular case in which bullet voting was considered. The term was used in The Journal of Politics (2007) by Jack H. Nagel, who named it after Aaron Burr, who initially tied with Thomas Jefferson for Electoral College votes in the United States presidential election of 1800.^{[16] [17]} According to Nagel, the electoral tie resulted from "a strategic tension built into approval voting, which forces two leaders appealing to the same voters to play a game of Chicken."

In a Burr scenario, a group of voters prefer two candidates (traditionally called Jefferson and Burr) from the same political party or faction (traditionally called the Republicans or Democratic-Republicans, not to be confused with the Republican Party). These candidates face a unified opposition (the supporters of Adams). This creates a dilemma:

• If a Republican voter supports both Jefferson and Burr, they are effectively casting no votes because they do not make a distinction between the top two candidates. If most Republicans support both, the election is also a near-tie, with the outcome being determined essentially by chance.
• On the other hand, if too many electors vote only for Jefferson or only for Burr, Adams will be elected.

The 1800 United States presidential election was conducted using a voting-rule similar to approval voting, though not quite identical. Each member of the Electoral College was allowed to vote for two candidates, with all votes being counted anonymously and simultaneously. The candidate with the most votes would be president, and the one with the second-most would be vice-president.

The Democratic-Republicans held a majority in the Electoral College that year, with 73 electors versus only 65 Federalists. Most of these electors were instructed to vote for both Jefferson and Burr, with the intention of securing both the Presidency and the Vice-Presidency for their party.

Ultimately the electors fell into the first trap, with all 73 Republicans (terrified by a possible Adams victory) supporting both Jefferson and Burr. The resulting tie nearly caused a constitutional crisis when the tiebreaking mechanism deadlocked as well.

Solutions

• Score voting lessens the Burr dilemma when some voters are either honest or willing to cooperate. In the real Burr election, honest electors might assign a score of 10/10 to Jefferson, 9/10 to Burr, and 0/10 to Adams; or they might all agree ahead of time to use this strategy. This strategy gives Jefferson 73 points and Burr 65.7 points. However, score voting systems can behave similarly to approval if voters act in a way that is individually strategic, suggesting score voting can only be a partial solution.
• Paired tournaments (i.e. Condorcet methods) elect candidates who win a pairwise majority vote. In the race above, Jefferson would win against Adams 73-65, while Jefferson would likely defeat Burr in a one-on-one election, allowing Jefferson to safely secure the presidency. However, Condorcet methods have their own advantages and disadvantages; for example, they can suffer from spoiler effects, which do not occur in honest approval and score voting.
• STAR voting is a compromise between score and the ranked Condorcet methods, which still maintains a "mostly score-based" system. Candidates give each candidate a score, and the top-two candidates face a runoff, where the candidate preferred by a majority wins.

Further reading

• Behind the Ballot Box: A Citizen's guide to voting systems, Douglas J Amy, 2000.
• Mathematics and Democracy: Recent advances in Voting Systems and Collective choice, Bruno Simeone and Friedrich Pukelsheim Editors, 2006.

Notes and References

1. https://patch.com/new-york/huntington/bp--bullet-voting Bullet Voting Explained
2. News: Drawing the Line . dead . https://web.archive.org/web/20170221150439/https://www.splcenter.org/20020213/drawing-line . 2017-02-21 . 2017-07-13 . Southern Poverty Law Center . en . 4. Anti-single-shot provisions: These provisions compel voters to cast a vote for every open seat, even if voters do not want to support more than one candidate. A voter who casts a vote for less than the entire number of seats open (a "full slate") will not have their ballot counted. Requiring minority voters to vote for a full slate dilutes their voting strength by preventing them from concentrating their support behind one candidate..
3. https://www.comoxvalleyrecord.com/opinion/editorial-to-plump-or-not-to-plump-your-vote/ EDITORIAL: To plump, or not to plump your vote
4. Web site: Later-No-Harm Criterion . 2024-02-02 . The Center for Election Science . en-US.
5. Graham-Squire . Adam . McCune . David . 2023-06-12 . An Examination of Ranked-Choice Voting in the United States, 2004–2022 . Representation . en . 1–19 . 2301.12075 . 10.1080/00344893.2023.2221689 . 0034-4893.
6. News: Does "Bullet Voting" Really Work? - Philadelphia Magazine. 2015-10-27. Philadelphia Magazine. 2017-07-12. en-US.
7. Web site: Ocean City Maryland News OC MD Newspapers Maryland Coast Dispatch » Merits Of Single-Shot Voting Questioned. 22 October 2008. mdcoastdispatch.com. en-US. 2017-07-13. Single-shot voting is essentially a tactic used by voters ... choosing only one candidate or a lesser amount of candidates than open seats..
8. Nagel . Jack H. . February 2007 . The Burr Dilemma in Approval Voting . . 69 . 1 . 43–58 . 10.1111/j.1468-2508.2007.00493.x . 10.1111/j.1468-2508.2007.00493.x . JSTOR.
9. Book: Nagel, Jack H. . Mathematics and Democracy. Studies in Choice and Welfare . 2006 . Springer . 978-3-540-35603-5 . Simeone . B. . Berlin, Heidelberg . 133–150 . A Strategic Problem in Approval Voting . 10.1007/3-540-35605-3_10 . Pukelsheim . F. . https://doi.org/10.1007/3-540-35605-3_10.
10. https://books.google.com/books?id=yKk6WluVMrsC&dq=anti-bullet%2Bvote&pg=PA449 Decision 1997: Constitutional Change in New York
11. Web site: Zawora . Deb Otis, Chris . 2021-08-16 . Rate of “Bullet Voting” Depends on Candidate Strength, Party Cues, and Other Factors . 2024-05-13 . FairVote . en.
12. Democracy in Divided Societies: Electoral Engineering for Conflict Management, Benjamin Reiley, 2001 p.145 ("But the Bucklin system was found to be defective, as it allowed a voter's second choice vote to help defeat a voter's first choice candidate. Under these circumstances, most voters refrained from giving second choices, and the intent of discovering which candidate was favored by the majority was thwarted.)"
13. Web site: 7 June 2020 . Approval Voting is Better Than Plurality Voting, Even in Multi-Winner Races .
14. Amy (2000) p.60 (At-large voting can discourage voters from supporting all the candidates they want to see on the council, a practice called bullet voting... This is a political predicament racial minorities find themselves. They must give up all of their other votes to have any hope of electing their first choice.)
15. Web site: Black candidate for Euclid school board to test new voting system . 2011-06-07 . https://web.archive.org/web/20110607170043/https://www.cleveland.com/news/plaindealer/index.ssf?%2Fbase%2Fcuyahoga%2F1250843460287750.xml&coll=2 . 2011-06-07 . live .
16. Nagel . Jack H. . February 2007 . The Burr Dilemma in Approval Voting . . 69 . 1 . 43–58 . 10.1111/j.1468-2508.2007.00493.x . 10.1111/j.1468-2508.2007.00493.x . JSTOR.
17. Book: Nagel, Jack H. . Mathematics and Democracy. Studies in Choice and Welfare . 2006 . Springer . 978-3-540-35603-5 . Simeone . B. . Berlin, Heidelberg . 133–150 . A Strategic Problem in Approval Voting . 10.1007/3-540-35605-3_10 . Pukelheim . F. . https://doi.org/10.1007/3-540-35605-3_10.